Parallax in the Lab

When you look at things from two
different points of view, nearby objects appear to shift with respect to more
distant ones. This is called parallax, and it's a basic tool for
measuring astronomical distances. The same technique can be used to measure
distances to objects on Earth.

Background Reading:Stars & Planets, p. 10 to 12 (Star distances).

Astronomers first used parallax to
measure distances within the Solar System in 1672, but living organisms have
been using parallax for several hundred million years - ever since the
evolution of the first animal with two eyes in its head. Two eyes are better
than one because they give you two slightly different views of the world; by
combining these views, your brain can estimate distances to nearby objects. The
parallax measurements we will make in this lab use a technique you have been
practicing since infancy. In some sense, you are already an expert at using
parallax to measure distances, but at the same time, you may have no idea how
your brain accomplishes this very useful trick.

JUDGING DISTANCE

A simple experiment illustrates the role
of binocular vision - that is, vision using two eyes - in judging
distance. First, close both eyes and lift one hand over your head. Have your
lab partner place a coin (or other small object) on the table within reach in
front of you. Now open both eyes and quickly lower your hand so that the
tip of your finger lands on the middle of the coin. You should have no trouble
doing this; try it a few times - with the coin in a different place each time -
to convince yourself that you can always place your finger more or less exactly
on top of the coin. (If you consistently miss the coin, you may not be
employing both eyes - get your vision checked!)

Now try the same thing again, but this
time, open only one eye (no peeking - cover your other eye to make
sure). You will probably have much more trouble putting your finger down on top
of the coin. Again, try this a few times with the coin in a different place
each time. About how often do you hit the coin? Do you tend to reach too far,
or not far enough? Try using your other eye - is it any better?

PARALLAX MEASUREMENT

This diagram is an overhead view showing
the geometry of a parallax measurement. Such a measurement requires
observations from two different places separated by a known distance. This
distance, the baseline, is represented by the symbol b. Pick a
fairly nearby target which you can view in front of a background much further
away (for example, you might use the pole of a streetlight as your target, with
the side of the valley as a background). For the first observation, line the
target up with some definite landmark in the background (for example, a rock on
the side of the valley). Now move to your second observation point, and use a
cross-staff to measure the angle between
your target and the background landmark. The distance D to your target
is

This formula is fairly easy to derive using
simple geometry. We will cover the derivation in class. The angle should
be measured in degrees. Note that it does not matter what units you use for b;
you will automatically get D in the same units!

An Example

The pictures below show how to make a
parallax measurement. For simplicity, I chose a fairly unexciting target - the
top of an electricity pole near my home, which I can view in front of the side
of a hill somewhat further away. As the background landmark, I used a
transformer on another electricity pole on the distant hillside. The first
picture just shows the overall situation.

Target
and background for a parallax measurement. The target (top of pole, on right)
and background landmark (transformer, on left) are marked by arrows.

To make the first observation, I moved
around to line up the target with background landmark, as shown below (on the
right). I used a pebble to mark the location of my first observation. I then
shifted to my left until the target and the background were no longer lined up,
as shown below (on the left). The distance to shift is arbitrary, as long as
the target and background landmark now appear comfortably separated from each
other. I used another pebble to mark the location of my second observation. The
baseline distance between the two pebbles was b = 45 inches.

Second
observation: target is visibly shifted with respect to background.

First
observation: target and background landmark are lined up with each other.

Now, from my second observation point, I
used a cross-staff to measure the angle between
the target and the background object. This is a little tricky, since it's hard
to keep the background, the target, and the ruler in focus at the same
time; the picture below shows that my camera also had some trouble focusing.
Nonetheless, even this fuzzy image is clear enough to show that the background
landmark falls at the 16.0 cm mark on the ruler, while the target falls at
about 16.8 cm. Thus the apparent separation between the target and the
background is 16.8 cm - 16.0 cm = 0.8 cm.
Since 1 cm on the ruler represents an angular separation of 1°, the angle
between the target and the background landmark is about = 0.8°.

Measurement
of parallax angle. The dotted lines show where the background landmark (left)
and target (right) fall along the cross-staff ruler.

Using b = 45 inches
and = 0.8°
in the formula above, I get D = 3200 inches = 270 feet. These
results are given to slightly better than one significant figure; the
measurement of could
be off by ±0.1°, so there's no point in trying to claim any higher level of
accuracy. The most serious source of error is my use of a background landmark
which is only a few times further away than the target. For example, if the
background is about five times further away than the target, the resulting
value of D will be about 20% too large.

PARALLAX EXPERIMENTS

In addition to the simple experiment on distance judging described above,
you will also make two measurements of distance using parallax. One measurement
will be performed during the lab; we will set up a suitable target and coach
everybody on the proper technique. The second measurement should be made later,
using a target and background that you chose. The key here is not just
to make a measurement - you will also have to make some choices, and explain why
you made those choices. At every step, your choices affect the accuracy of your
result, so think carefully when choosing.

Note: keep in mind that binocular parallax is not the only way your
brain judges distances. For example, if you move your head around, nearby objects
appear to shift more than distant ones; this is another form of parallax which
can help you judge distances using only one eye. The apparent sizes of objects,
and the amounts of light they reflect, also provide some clues to their
distances. All of these tricks, which are `built in' to our brains, are also
used to measure distances in astronomy.